|
| |
|
|
A178514
|
|
Triangle read by rows: T(n,k) is the number of derangements of {1,2,...,n} having genus k (see first comment for definition of genus).
|
|
5
|
|
|
|
0, 1, 0, 1, 1, 0, 3, 6, 0, 0, 6, 30, 8, 0, 0, 15, 130, 120, 0, 0, 0, 36, 525, 1113, 180, 0, 0, 0, 91, 2016, 8078, 4648, 0, 0, 0, 0, 232, 7476, 50316, 67408, 8064, 0, 0, 0, 0, 603, 27000, 281862, 719640, 305856, 0, 0, 0, 0, 0, 1585, 95535, 1459920, 6298930, 6223800, 604800, 0
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,7
|
|
|
COMMENTS
|
The genus g(p) of a permutation p of {1,2,...,n} is defined by g(p) = (1/2)(n + 1 - z(p) - z(cp')), where p' is the inverse permutation of p, c = 234...n1 = (1,2,...,n), and z(q) is the number of cycles of the permutation q.
The sum of the entries in row n is A000166(n) (the derangement numbers).
The number of entries in row n is ceiling(n/2).
T(n,0) = A005043(n) (the Riordan numbers).
|
|
|
LINKS
|
|
|
|
EXAMPLE
|
T(3,1)=1 because 312 is the only derangement of {1,2,3} with genus 1. Indeed, we have p=312=(132), cp'=231*231=312=(132) and so g(p) = (1/2)*(3+1-1-1) = 1, while for the other derangement of {1,2,3}, q=231=(123), we have cq'=231*312=123=(1)(2)(3) and so g(q) = (1/2)*(3+1-1-3) = 0.
Triangle starts:
[ 1] 0,
[ 2] 1, 0,
[ 3] 1, 1, 0,
[ 4] 3, 6, 0, 0,
[ 5] 6, 30, 8, 0, 0,
[ 6] 15, 130, 120, 0, 0, 0,
[ 7] 36, 525, 1113, 180, 0, 0, 0,
[ 8] 91, 2016, 8078, 4648, 0, 0, 0, 0,
[ 9] 232, 7476, 50316, 67408, 8064, 0, 0, 0, 0,
[10] 603, 27000, 281862, 719640, 305856, 0, 0, 0, 0, 0,
[11] 1585, 95535, 1459920, 6298930, 6223800, 604800, 0, 0, 0, 0, 0,
[12] 4213, 332530, 7117902, 47851540, 90052336, 30856320, 0, 0, 0, 0, 0, 0,
...
|
|
|
MAPLE
|
n := 7: with(combinat): P := permute(n): inv := proc (p) local j, q: for j to nops(p) do q[p[j]] := j end do: [seq(q[i], i = 1 .. nops(p))] end proc: nrfp := proc (p) local ct, j: ct := 0: for j to nops(p) do if p[j] = j then ct := ct+1 else end if end do: ct end proc: nrcyc := proc (p) local pc: pc := convert(p, disjcyc): nops(pc)+nrfp(p) end proc: b := proc (p) local c: c := [seq(i+1, i = 1 .. nops(p)-1), 1]: [seq(c[p[j]], j = 1 .. nops(p))] end proc: gen := proc (p) options operator, arrow: (1/2)*nops(p)+1/2-(1/2)*nrcyc(p)-(1/2)*nrcyc(b(inv(p))) end proc: DER := {}: for i to factorial(n) do if nrfp(P[i]) = 0 then DER := `union`(DER, {P[i]}) else end if end do: f[n] := sort(add(t^gen(DER[j]), j = 1 .. nops(DER))): seq(coeff(f[n], t, j), j = 0 .. ceil((1/2)*n)-1); # yields the entries of the specified row n
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
|
|
|
STATUS
|
approved
|
| |
|
|
